1. Field of the Invention
The present invention relates generally to the regularized solution of ill-conditioned problems and more particularly to ill-posed inverse problems, such as those found in the field of electrocardiography. An exemplary problem is the generation of a mapping of electrical potentials at the surface of the heart, based on measured potentials at positions that are not in contact with the heart surface.
2. Related Art
Every year in the United States alone, more than 200,000 patients die suddenly, due primarily to specific heart rhythm disorders that are caused by abnormal, rapid beats. Cardiac arrhythmia is one of the most common disorders in clinical practice. Current drug therapy for these irregular heart rhythms is often ineffective and, at times, can further complicate the rhythm abnormalities (including increased mortality). (For example, Pharmacologic therapy is generally not effective in the management of tachyarrhythmias.Super]) Radiofrequency ablation, a relatively new, minimally invasive, non-drug therapy has been shown to be relatively safe and effective in treating selected rhythm disorders; catheters inside the heart are used to deliver a energy to abnormal regions abolishing the cause of irregular beats. However, ablation of rhythm abnormalities that are brief, chaotic or hemodynamically unstable can be difficult given the limitations of the tools that are available to pinpoint the cause of these irregular beats. Radiofrequency (RF) catheter ablation has revolutionized the treatment of heart rhythm disorders, especially those for which the mechanisms are well understood, resulting in successful treatment. (For example, RF catheter ablation has revolutionized the treatment of supraventricular tachycardias, i.e., Wolff-Parkinson-White syndrome and atrioventricular nodal reentrant tachycardia, and atrial flutter.) However, for some disease the mechanism is not clear. For example, the mechanism is not clear in the cases of ventricular tachyarrythmia and atrial fibrillation. This is largely due to the inability to properly map the hearts activity using current mapping methods. To solve the problems associated with drug therapy and to advance alternative catheter-based therapies for eliminating heart rhythm abnormalities, improved methods are needed to map the electrical activity of the heart. The mapping of the heart s electrical activity can enable the physician to better localize the origin of irregular beats, to identify their causes, and to deliver the proper therapy, quickly. The solution of the discrete, ill-posed inverse problem of electrocardiography (the generating of a mapping of the hearts electrical activity based on measured potentials within the heart) is therefore needed.
The solution of the electrocardiographic inverse problem can aid in the diagnosis and treatment of disease manifesting itself in the hearts electrical activity. This must be done with care because of the ill-posed nature of the problem, as reflected in the ill-conditioning of the coefficient matrix (transfer function) that models the physical system. (Ill-conditioned, as used herein, refers to the fact that the system is near-numerically singular.) Inherent in the solution of the inverse problem using computer-based methods is the amplification of the measurement errors—often resulting in wildly varying solutions that are orders of magnitude greater than the heart is capable of producing. (For example, solutions in the millions of volts rather than in the tens of millivolts.) A simplified explanation of this problem is, for example, that the extraction of information from the measurements involves an amplification of parts of the signal to counter losses. In the process, any noise which is present is also amplified and, in fact, overwhelms the desired information. Several methods are currently used to mitigate the ill effects due to the poor conditioning and noisy measurements. Each method has its strengths and shortcomings. Some of these methods are mentioned later.
Typically, an instantaneous solution to the steady-state problem has been found with measurements made at a specific time. However, this problem has a temporal nature. Consider that a cardiac cell s activation stimulates neighboring cells and that this activation wave front travels outward through the myocardium (along the cardiac fibers). This temporal information is not part of the governing electrostatic equations (e.g., Maxwell s equations) and is not typically incorporated into solutions of inverse problems such as the electrocardiography inverse problem.
Just as there is a need to provide improved means for solving the inverse problem of electrocardiography described above, there is a need to provide improved means for solving other types of inverse problems. These problems may involve potential flow, heat transfer, groundwater flow, laminar fluid flow, etc. It is often the case that the desired information/signals cannot be observed or measured directly. Alternatively, making indirect measurements may be necessary in order to avoid some difficulty related to direct measurements. The indirect, measurable information/signals for these problems then require processing in order to extract/reconstruct the desired information/signals. In the electrocardiography example above, it is desired to map the electrical activity of the heart during disease related episodes when the heart s pumping action is compromised, but it is difficult to directly measure this electrical activity quickly (without grave risk to the patient). It is therefore desirable to be able to measure electrical potentials within one of the hearts chambers (thereby avoiding contact with the heart muscle itself) and to reconstruct the mapping of the heart s electrical activity from this information.